These chapters come from the text: Hildebrand, Francis. Advanced Calculus for Applications. 2nd ed. Englewood Cliffs: Prentice Hall, March 31, 1976. ISBN: 0130111899.
| LEC # | TOPICS | READINGS |
|---|---|---|
| 1-2 | Number Systems and Algebra of Complex Numbers Elementary Complex Functions, Part 1 | Algebra of Complex Numbers (10.1) Elementary Complex Functions (10.2, 10.3) |
| 3-4 | Elementary Complex Functions, Part 2 Branch Points and Branch Cuts | Analytic Functions (10.3, 10.4) Convergence of Power Series: Examples (4.1, 4.2) Line Integrals, Cauchy's Formula (10.5, 10.6) |
| 5-6 | Analytic Functions Complex Integrals | Taylor Series, Laurent Series (10.7, 10.8) Singularities (10.9) |
| 7-8 | Cauchy's Formula, Properties of Analytic Functions Taylor Series, Laurent Series | Singularities at Infinity (10.10, 10.11) The Residue Theorem (10.12) |
| 9-10 | Laurent Series (cont.) Properties of Laurent Series, Singularities | Evaluation of Real Definite Integrals (10.13) Handout 1 on Overview of Evaluation of (Real) Definite Integrals (PDF) Limiting Contours (10.14) Indented Contours (10.15) |
| 11-12 | Singularities (cont.) Residue Theorem | Indented Contours (10.15) Integrals Involving Branch Points (10.16) |
| 14-15 | Evaluation of Real Definite Integrals, Case I Evaluation of Real Definite Integrals, Case II | Singular Points of Linear Second-Order ODEs (4.3) The Method of Frobenius (4.4) Handout 2 on An Overview of the Fobenius Method (PDF) |
| 16-17 | Evaluation of Real Definite Integrals, Case III Evaluation of Real Definite Integrals, Case IV | The Method of Frobenius - Exceptional Cases (4.4, 4.5, 4.6) |
| 18-19 | Theorems for Contour Integration Series and Convergence | Bessel Functions (4.8) Properties of Bessel Functions (4.9) |
| 20-21 | Ordinary Differential Equations Singular Points of Linear Second-Order ODEs | Differential Equations Satisfied by Bessel Functions (4.10) Legendre Functions (4.12) |
| 22-23 | Frobenius Method Frobenius Method - Examples | Introduction to Boundary Value Problems (5.1) The Rotating String, The Rotating Shaft (5.2, 5.3) |
| 24-25 | Frobenius Method (cont.) and a "particular type" of ODE Bessel Functions | Orthogonality of Characteristic Functions (5.6) Expansions in Series of Orthogonal Functions (5.7) |
| 26-27 | Properties of Bessel Functions Modified Bessel Functions | Boundary Value Problems for Nonhomogeneous PDEs (5.8) Fourier Series (5.10) Complete Fourier Series (5.11) |
| 29-30 | Differential Equations Satisfied by Bessel Functions Introduction to Boundary-Value Problems | Fourier-Bessel Series (5.13) Legendre Series (5.14) The Fourier Integral (5.15) |